This relaxed subproblem \({\hat{\cal P}_{ \text{ passive-relaxed}}}\)  is convex, which is now similar to the subproblem \(\hat{\cal P}_{\rm passive}\)  in (REF ). Thus, \({\mathbf {\tilde{\Theta }} }^{\rm opt}\)  can be directly solved by the standard convex tools such as CVX [1]} as well. Then, based on the proximity principle, we can map the solved \({\mathbf {\tilde{\Theta }} }^{\rm opt}\)  to the elements in \({\cal F}_2\)  or \({\cal F}_3\)  by a quantization operation, which is common in existing works [2]}, [3]}, [4]}, and can be written as
\(\begin{aligned}\!\!\left\lbrace  \begin{array}{l}\left| {\theta _{r,j}^{\rm sub}} \right| = 1,\\\angle \theta _{r,j}^{\rm sub}{\rm { = }}\mathop {{\rm {argmin}}}\limits _{\phi \in {\cal F}_2~{\rm or}~{\cal F}_3} \left| {\angle \tilde{\theta }_{r,j}^{\rm opt} - \angle \phi } \right|,\end{array} \right.~~\forall r\in {{\cal R}},\forall j\in {{\cal N}},\end{aligned}\) 
